Answer:
= 1087.5
Step-by-step explanation: To evaluate the triple integral, first an equation of a plane is needed, since the tetrahedon is a geometric form that occupies a 3 dimensional plane. The region of the integral is in the attachment.
An equation of a plane is found with a point and a normal vector. <u>Normal</u> <u>vector</u> is a perpendicular vector on the plane.
Given the points, determine the vectors:
P = (5,0,0); Q = (0,9,0); R = (0,0,4)
vector PQ = (5,0,0) - (0,9,0) = (5,-9,0)
vector QR = (0,9,0) - (0,0,4) = (0,9,-4)
Knowing that cross product of two vectors will be perpendicular to these vectors, you can use the cross product as normal vector:
n = PQ × QR = ![\left[\begin{array}{ccc}i&j&k\\5&-9&0\\0&9&-4\end{array}\right]](https://tex.z-dn.net/?f=%5Cleft%5B%5Cbegin%7Barray%7D%7Bccc%7Di%26j%26k%5C%5C5%26-9%260%5C%5C0%269%26-4%5Cend%7Barray%7D%5Cright%5D)
![\left[\begin{array}{ccc}i&j\\5&-9\\0&9\end{array}\right]](https://tex.z-dn.net/?f=%5Cleft%5B%5Cbegin%7Barray%7D%7Bccc%7Di%26j%5C%5C5%26-9%5C%5C0%269%5Cend%7Barray%7D%5Cright%5D)
n = 36i + 0j + 45k - (0k + 0i - 20j)
n = 36i + 20j + 45k
Equation of a plane is generally given by:

Then, replacing with point P and normal vector n:

The equation is: 36x + 20y + 45z - 180 = 0
Second, in evaluating the triple integral, set limits:
In terms of z:

When z = 0:

When z=0 and y=0:
x = 5
Then, triple integral is:

Calculating:





![\frac{1}{45} [30375-60750+118462.5-39150]](https://tex.z-dn.net/?f=%5Cfrac%7B1%7D%7B45%7D%20%5B30375-60750%2B118462.5-39150%5D)
= 1087.5
<u>The volume of the tetrahedon is 1087.5 cubic units.</u>